By the Riemann–Lebesgue lemma [1]}, \(\limsup _{|t|\rightarrow \infty }|\varphi _{1,ac}(t)]|=0\)  and by [2]}, \(\limsup _{|t|\rightarrow \infty }|\varphi _{1,d}(t)]|=1\)  and \(\limsup _{|t|\rightarrow \infty }|\varphi _{1,sc}(t)]|\in [0,1]\)  (the \(\limsup \)  can in fact take all values in the interval).
That is why non-singularity implies eq.cramer. If \(\limsup _{|t|\rightarrow \infty }|\varphi _{1,sc}(t)]|=1\)  (for examples see [3]} or [2]}) or \(p_{sc}=0\) , non-singularity and eq.cramer are equivalent and both imply that the distribution of \(X_1\)  is not purely discrete.
